on the structure of continuous functions of several variables … · continuous functions of...
TRANSCRIPT
ON THE STRUCTURE OF CONTINUOUS FUNCTIONSOF SEVERAL VARIABLES
BY
DAVID A. SPRECHER*1)
1. Introduction and statement of the results. This paper deals with the
representations of arbitrary real continuous functions, of any number of
variables, as finite sums of real continuous functions of one variable. The
first such representation theorem was proved by A. N. Kolmogorov [5]
in 1957. Our Theorem 1 is a refinement of Kolmogorov's result, and
contains the latter as a special case; it, in turn, is a special case of Theo-
rem 2 below.
Let S3n denote the cartesian product
^n= n -^
of closed unit intervals, the interval J^, being laid on the pth coordinate
axis of a rectangular coordinate system in ra-dimensional Euclidean space,
3ên; designate points in 38n by x:
x = (*i» • • •, x„) ;
3è will denote the real line and & the unit interval thereof. The image of
a set A C @n under a mapping <f> is designated by <bA :
<bA = {*(x)| xEA\.
Theorem 1. For each integer N s 2, (Aere exists a real, monotonie in-
creasing function, i(x) ELip[ln2/ln(22V-r- 2)], $& = &, dependent on N,
and having the following property(2):
For each preassigned number 5 > 0 iAere is a rational number t, 0 < t 5¡ ô,
such that for 2 z% n zZ N every real continuous function of n variables, f(x),
defined on h3n, has a representation as
(1.1) /(x)= Z x\ £ WixP + eq)+q
Presented to the Society, January 26, 1965 under the title On the representations of continu-
ous functions of several variables; received by the editors December 12, 1963 and, in revised
form, March 16, 1964.
( ) Part of this research was supported by the United States Air Force through the AFOSR
under Contract No. AF 49 (638) -590, and by the National Science Foundation under Grant
No. NSF-16003; part of the results herein are contained in the author's Ph.D. thesis
(University of Maryland, 1963).
( ) iA(i) belongs to class Lip[a] if there are constants c and a, 0 < a g 1, for which
I t(x) — t(y) I S | x — y | " for all points, x and y, in the domain of ^.
■
340
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CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 341
where the function \ is real and continuous, and X a constant independent of f.
With an obvious modification in requirement (4.7) and the proof of
Lemma 2 below, one can prove Theorem 1 for representations of the form
(1.2) /(x)= Z x\ Z \^ixp + tq)],0SqS2n L.1SPS" J
X again being a constant independent of /. We mention in passing that the
powers of the single number X can be replaced with any set of n real numbers
Xp, 1 ^ p ^ n, which do not satisfy any homogeneous linear equation with
integral coefficients: this is readily verified by analysing the proof of Lemma
2 below.
Kolmogorov [5] proved the existence of representations of the form
(1.3) /(x)= Z x,!" £ M*p)~|;0£qS2n |_lSp£n J
his 2n2 + n functions ^M are here replaced by the single function \f/ plus
constants; the x? are replaced by a single x, a simplification that follows,
as was first noted by G. G. Lorentz [6], from refining one of Kolmogorov's
own constructions; the intervals in [5, Lemma 2(3)] are modified so as to
be disjoint for variable q.
The specific constructions in our paper are much simpler than the cor-
responding ones suggested in [5], because the construction of ^ is independent
of the parameter q, which appears in the remaining constructions only as
an additive constant; they do, however, share their topological properties.
In particular, Lemmas 1-2 above imply the corresponding Lemmas 1-2 in
[5], but not conversely. Kolmogorov's result depends on three lemmas,
stated in [5] without proof. Elegant proofs of these have been given by
J. Kim in his master's thesis (University of Maryland, 1960, unpublished) ;
subsequent proofs of Kolmogorov's theorem were given by Arnol'd [2],
Lorentz [6] and Tihomirov [9].
For the case n = 3, Theorem 1 contains as a special case a result of
Arnol'd [l], who obtained a corresponding representation with nine sum-
mands and considerably more complicated arguments (see also [5]).
Theorem 1 depends on the lemmas stated below. Since our results are
valid for any preassigned integer N g n, we shall assume that N = n. It
will be evident from the construction of the function \p that we can do so
without any loss of generality. Throughout this paper, r will designate the
set of natural numbers; p will be an index with domain 1 á P á "■ The
diameter of a set A G &" is the number
5(A) = sup |x-y|,*,l£A
| x — y | denoting the Euclidean distance between the points in question.
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342 D. A. SPRECHER [March
Let 7 be a constant, y > 1. For each A£r, Sk will denote a certain
finite family of pairwise disjoint, nonempty, closed sets in 3èn, with di-
ameters not exceeding 7~*_1; the members of Sk will be designated by
Skii), where i is an index with a certain finite domain A*, for each A, Ak
depending on the particular choice of sets involved.
Let v be a fixed vector, v = ivlt ■••,y„); let q and A be fixed. For each
i E Ak, designate by SUi) the set S°(¿) displaced along v by a distance q:
(1.4) SUi) = {x + q . v| x G S&i), i E A*} ;
denote the family of sets (1.4) by S\.
Lemma 1. TAe family Sk and the vector v can be so determined, that the
union of families SI, 0 zZ q z% n + m, covers Sa"1 m + 1 times for each A.
Furthermore, the members of each family can be so labeled that, for each i = ¿0
in Ak, A being held fixed, there is a nonempty intersection of sets S?(¿):
(1.5) Skii)= D S2(i,)*0,OSqitoi
where ix, ••■,i2n are uniquely determined numbers in Ak, and i = (¿0, ••-,¿2™).
It is the special structure of the families SI that permits us to prove
Theorem 1 with a single function ^.
Lemma 2. Let a number & > 0 be given; select a natural number k = k0 so
large, that e = (7 — 1)_17_*°^*- There exists a real, monotonie increasing
function, \f/ix), dependent on n, such that xf/té1 = &, and a real constant, X,
such that the functions
(1.6) *'(x)= Z Xp*ixp+(q)+q,lápSn
0 iZq z%2n, have the property
(1.7) gqSM) ng'Siii') = 0
for each k, unless r = q and i' = i. Moreover, the function $ can be constructed
to belong to class Lip[ln2/ln(2rc + 2)].
A function / separates the distinct points x and y if their corresponding
images under / are distinct. For an arbitrary continuous function, fix),
imagine a representation such as
(1.8) /(x)= Z x[h"ix)},OiqSm
the functions x and A' being continuous, and m arbitrary. Because the
right side of (1.8) is invariant under all permutations of the functions A',
it follows that it separates the points x^y only if for no permutation
(a0, • • •, am) of the tuple (0,1, • • •, m) is the set (A°(x), • • •, Am(x)) a permuta-
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 343
tion of the set (A"°(y), •••,A°m(y)). Clearly, this requirement is necessary
for all representations of continuous functions in the form (1.8). We
prove elsewhere [11] that it is not sufficient. We prove, namely, that
there are continuous functions hq which separate all points of &n, but for
which not every continuous function fix), x G E", can be represented in
the form (1.8).
The functions gq of Lemma 2 separate the sets S\i for each A: such func-
tions have the property stated in
Corollary 1. Any set of 2n + 1 functions satisfying condition (1.7) sepa-
rate all points of Sé"1.
Theorem 2. Let {A'(x)} be any set of n + m continuous functions with
domain En which, in addition to property (1.7), separates all points of Wn.
Then every continuous function, fix), defined on W1, can be represented in
the form (1.8) with n + m superpositions of a continuous function \, where
»»21.
It is requirement (1.7) for variable i that restricts the smoothness of
4/; one can readily show that a function satisfying that condition cannot
belong to class Lip[l]. Therefore, in order to determine the best possible
f for which Theorem 1 can be proved, one has to know whether or not
condition (1.7) is necessary; we do not know this to be the case.
The author wishes to express his gratitude to A. Douglis for his invaluable
suggestions and criticism; his insight brought into focus many of the results
presented here.
2. Concomitant remarks. The existence of continuous functions of several
variables that are not finite superpositions of continuous functions of a
lesser number of variables was conjectured by Hilbert in the thirteenth
problem of his famous Paris lecture of 1900 [3]. Studying the roots of
polynomial equations, he concluded that there are proper functions of more
than one variable. Hilbert conjectured, namely, that the roots of the equation
x7 + ax3 + bx2 + ex + 1 = 0
are functions of its coefficients that cannot be represented as finite super-
positions of continuous functions of only two arguments. This conjecture
was refuted in 1957 by Arnol'd [l] and Kolmogorov [5]; Kolmogorov, in
fact, proved the general representation theorem quoted in the introduction.
In connection with his thirteenth problem, Hilbert also raised questions
concerning the classification of continuous functions of n variables according
to their representability in terms of continuous functions of m variables,
m s; n, of a given class (say, analytic, algebraic, etc.) [4]. Owing to Theo-
rem 1 and Theorem 2, such a classification fails when we wish to classify
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344 D.A.SPRECHER [March
continuous functions of n variables according to their representability in
terms of continuous functions of m variables, for then m = 1. In this case,
however, one could use as classification index the least number of summands
in (1.8) for fixed functions A'.
We mention in passing that every continuous function of n variables,
fix), can be represented as
(2.1) fix)=g\ Z *p(*p)1,LlSpán J
where the functions \(/p are monotonie increasing and independent of /,
if the function g is permitted to be discontinuous [8].
Extensive work along the lines suggested by Hilbert in [4] was done
by A. G. Vituskin; we wish to single out one significant result [10]:
We consider those functions of n variables having all partial derivatives
of orders ^p, and whose derivatives of order p belong to class Lip [a]: we
associate with such functions the index p = n/ip + a), where we assume
that p + a ^ 1. Vituskin's theorem states that not all continuous functions
with index p can be represented with superpositions of functions with index
Mo, if M < Mo-
lt is not yet known how to determine the class, say, of analytic func-
tions of several variables that can be represented as finite superpositions
of analytic functions of one variable; similar questions arise for other classes
of functions. In particular, it would be of interest to determine whether
or not a Lipschitz continuity of / in (1.1) implies a Lipschitz continuity of
x; in general, we do not expect / and x to belong to the same exponent a.
In this connection, also the following question arises: In what way is the
smoothness of x and ^ in (1.1) dependent on the number of superpositions?
The number of summands in representations of the form (1.1) can be
reduced when one is willing to sacrifice some of the properties of our re-
sults (see, for example, (2.1) above). The function /(x,y) = xy can be
written as
fix.y) = xxix + y) + x2(* + ( - y)),
where xi = £2/4 and x2= — |2/4: here, however, the arguments involved,
x + y and x+ ( — y), depend on /. Arnol'd [l] obtained, for n = 3, repre-
sentations with only three superpositions of two variables each, but there
the functions vary over a space that is considerably more complicated
than an interval; they take on values on the cartesian product on an inter-
val and a tree.
3. Proof of Theorem 2. Of the families SI we need in this section only
their covering property and (1.5); the desired function will be obtained as
a limit, x = fim^cXr, of a sequence of polygonal arcs, {xr), to be deter-
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 345
mined by induction on r, simultaneously with a subsequence, {Ar}, of r.
Except for a slight modification, the construction that follows is that of
Kolmogorov, but the estimates in the convergence proof are new.
For each A, denote by Sk the union
(3.1) Sk = U S*(i)
(see (1.5)). Clearly, if the (fixed) function Xr-i(£) is continuous for lE&ë,
then the function
(3.2) fr-Ax)= Z Xr-1[A«(X)]
is continuous for xG^". Defining xo(!) — 0, so that /0(x) = 0, we set with
pr= SUP |/(X) - fAx),
(3.3) KvT = sup I fix) - frix) | ;
clearly, pr z% v„ and, in particular, p0 = "o-
The numbers pr-X and vr_x are determined by /r_i(x) and Ar_i; since the
inequality 5(Sg(¿)) zi 7"*"1 is uniform in ¿, we can select kr so large, that
Pr_i= sup f sup |[/(i)-/,-i(x)]-[/(y)-/r-i(y)]|}u slr\ x,,esL(¿) )
OSqSn+m
(3.4)1
-ñ+m + l""1'
Let A0=l; by definition, xo —0; assuming that the continuous function
Xr-i and number kr_x are already fixed; we select kr such that (3.4) holds.
In the sets Skrii) we fix arbitrary points, xkri, for each iq E Akr; we assume
tacitly that all these points belong to Sa"1.
On every interval A'S^(i) define the function Xr(£) by the equation
(3.5) Xrit) = Xr-l(l) + „ , I, . [f(xkrd - fr.XiXkrd },n -+- m -\-1
this being possible because the intervals under consideration are all mutually
exclusive. For the specified domain of £, xr(£) satisfies the inequality
(3-6) I XÁ& - Xr-ÁÚ I á . I , 1 fir-Ln -j- m -4-1
On the complement of the intervals A'iSHr(i) complete the definition of xAO
arbitrarily, but so that it remains subject to inequality (3.6) and continu-
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346 D. A. SPRECHER [March
ous, a process justified by the Tietze-Urysohn extension theorem.
We shall now estimate the difference, / — fr, at arbitrary points in Skr
and En, respectively: these estimates are possible because the functions
A'(x) together separate all points of Sfn. The first estimate will enable us
to show that the sequence x<-(£) converges uniformly to a continuous limit
function; the estimate over E" will yield Theorem 2.
Owing to (3.2) we may write
(3.7) fix) - fAx) = fix) - fr.Ax) - Z [Xrih") - Xr-Ah")],OSqSn+m
where A' = A'(x). For arbitrary points, x, in S^, each summand, Xr — Xr-u
can be represented by (3.5) for some vector i; with the help of (3.4), we
find that the right side of (3.7) does not exceed pr-X/in + m + 1) in ab-
solute value so that, in particular, pr ;£ pr_x/in + m + 1). Hence,
(3.8) pr è in + m + 1) ~rm = in + m + 1) 'W,
applied to (3.6), the last inequality shows that
(3.9) | xÁt) - Xr-AH) I < in + m + 1) "V0,
this inequality being valid for all points £ G &&. It follows that the sequence
Xr converges uniformly to a continuous limit function, x, as r—> oo.
To justify representation (1.8), we shall show that er—> °° as r—> <*>. Con-
sider equation (3.7) for an arbitrary point, iG-^" Because of the prop-
erties of the families SI, there are at least m + 1 values, q', of q, for which
x G <Sír(i) for some m + 1 vectors i. For these values of o we represent the
difference, Xr— Xr-u with the help of (3.5); in particular, we have:
(3.10) Xr(í) - Xr-l(í) = „.I,, [/« - /r-l(x) ] - «,n + m + 1
where, owing to (3.5) and (3.9),
(3.11) a^in + m<+ 1) "V-i èin + m + 1) "V-i Sin + m + 1) ~r~V
Substituting (3.10) into (3.7), we obtain
fix) - fAx) = , 1,1 [fix) - fr^ix)]- a+Z MA«) - Xr-l(A')].n + m + 1 ,^,<
The n summands are estimated with the help of (3.9) which, together with
(3.3) and (3.11) shows that
|/(x)-/r(x)| ^ nin + m + 1)\_,
+ [min + m + I)'1 + n]in + m + 1) ~rp0.
Because this inequality is valid at each point of &", it holds also for vr,
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 347
the supremum of the difference | / — fr\ ; an iteration leads to the inequality
"r- ( (n + m + IV + L (n + m + iy + in + m + Dr_1J n - if "°
as r—> oo. This completes the proof of Theorem 2.
4. Construction of the function ^(x). We shall construct in this section
the monotonie increasing function ^(x), for xE^ë, belonging to class
Lip[ln2/ln(2ra + 2)]. This function will be determined by a correspondence
between certain point-sets, generated from suitably selected systems of
nested intervals. Throughout this section, i will be an index with domain
I"-{0,1,2,...}.Let 7 denote a fixed integer, y ^ 2n + 2; for each natural number A
define the numbers
, , e*(0 = 17"*,(4.1)
* y'2 ~k
7-1
with corresponding intervals
(4.2) Ek(i) = [eluek + ôk],
where ek = ekii). As ¿ varies, with A held fixed, these intervals are separated
by gaps of width (7 —1)_17~*; for increasing A they have intersection
properties as follows:
(4.3) E„ii) n Ek+Xii') * 0
if and only if
(4.4) i' = yiJrt i0zitz%y-2).
Furthermore, for each ¿,
e*(») =ek+xiyi),
(4.5)c*(0 + àk = e*+i(7i + 7 - 2) + ôk+x
(see Figure 1); It follows that the Ekii) are nested or disjoint:
(4.6) Ekii) D Ek+Xii')
whenever (4.3) is valid.
As i varies over r", and as t admits values between 0 and 7 — 1, the
right side of (4.4) defines a single valued mapping of r" onto itself. Thus,
each element of I" is completely determined by the pair (¿, t), in a one-to-
one manner, and conversely. Only for t = y — 1 is the intersection (4.3)
empty (see Figure 1).
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348 D. A. SPRECHER [March
Ek+iü') r0^íáT-2 % ir-OátáT-2 >,
ekH) Ek{i) «*(»)+** Ek{i+1)
Figure 1
We construct by induction a sequence of natural numbers, {ßk}, as follows:
Let X > 0 be a fixed real number which is not a root of any polynomial
equation of degree less than n with integral coefficients. For each A, con-
sider the domain
&k=\-y\ ...,-i,o,i,. ••,7"*};
let hx,---,h„ designate arbitrary values in ^*; let Hk denote the set of
all nonzero vectors, (Ai, • • •, A„), with components in &k.
Let 0i = 1; suppose ßk already determined for A ̂ 1, and select ßk+x such
that
(4.7) 7-"*+i<7-i*-1minH*
£ hpXpISPSB
this is clearly possible, since the right side does not vanish. It is convenient
to let X be an algebraic number of degree n; applying [7, Theorem 3] to the
right side of (4.7), one can easily show that it exceeds -y-""*-0, where c is
a constant depending only on X and n. Thus, (4.7) will be satisfied if we
take ßk+x ̂ nßk + c; an iteration shows that
(4.8) A+lS(I + _^_)n.__^_.
Now define the numbers
hkij) = jy-\ e* = (7 - 2) • £ 7-'*+'(4.9) ,er
for JEl", and corresponding intervals
(4.10) Hkij) = [hk,hk+ik],
where hh = hkij), laid on a coordinate axis perpendicular to 3$. We ob-
serve that, for fixed A, consecutive intervals are separated by gaps of width
y~"k — ek, and one verifies readily that
(4.11) Hkij)C)Hk+yiJ')^0
if and only if
(4.12) y = ;/*+!-«* + s i0èsziy-2);
moreover,
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 349
(4.13) HAJ)DHk+xij')
whenever (4.11) holds. Thus, for increasing A, these intervals have inter-
section properties similar to those of the intervals EAi). For y — 1 ^ s
^ yfik+i-Pk _ i the intersection (4.11) is empty.
For each A, we relate certain of the intervals Hkij) to the EAi); the
association is done inductively, as follows:
Lemma (i). To eacA value of the index i, A being fixed, we can associate a
value, jk, of j, different jk to different i, such that
(a) For each A the association is monotonie increasing;
(b) The Hkijk) will be subject to condition (4.13) for the values j = jk and
j' =Jk+u for each A;
(c) TAe widths of the gaps between the intervals Hkijk) diminish to zero,
uniformly in jk, as A —■» œ.
This interval correspondence will lead to a point-correspondence between
the set of points that belong to infinitely many EAi), and the analogous
set, consisting of all points belonging to infinitely many Hkijk). The two
sets thereby related being dense (see below), this correspondence will enable
us suitably to define \f/.
Proof. Set jx = í ; for some value of A ̂ 1, suppose jk already determined
as a function of i; define jk+x = jk+Ai,0 as
, N . (jkyßk+1~ßk+t iO^t^y-2),
(4.14) jk+x = 1X < i • [jk+iH, 7 - 2) + 7*+1(¿ +1,0)]) (í = y - 1),
the brackets ( ) indicating the integral part of the enclosed quotient.
For fixed A, this correspondence is increasing, as demanded in (a). There-
by, by induction, a value./* is assigned to every pair (¿, t), where i ET',
and 0 á t Sí 7 — 1; in view of the one-to-one relation between these pairs
and all numbers of r", it follows that the association (4.14) assigns a value,
jk, to each number, *', where i' = y i + t, t admitting values between 0 and
y — 1 as described (see Figure 2). Also, owing to this correspondence, both
(4.3) and (4.11) are valid if and only if 0 ^ t ^ y — 2; this proves the
assertion made in (b). A simple calculation shows that the widest gap be-
tween any two intervals Hk(Jk) is bounded by 2~*-1, thus establishing (c).
Hk+Xijk+Ai, y - 2)) Hk+Ajk+Ai, y - D) Hk+Ajh+Ai + l, 0))
0^fg7-2) í=7-l| (OgígY-2
■ ■ ... ■ m oo • • • b o b b B ■ • • • a a o • • • ■ ■ ■ • ' • m o
HAJAi)) HAJAi + D)
Figure 2
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350 D. A. SPRECHER [March
It now follows that for fixed A, but any integer r > k, Hkijk) D HXjr)
whenever Ekii) DEr(i'), and conversely, where jk = jkii) and jr = jrii');
this is easily verified from (4.14) and the fact that (4.5), together with
the analogous property for the Hkijk), are true for each A. Thus, to every
nonempty infinite intersection of intervals Ekii) there corresponds a non-
empty infinite intersection of intervals Hkijk), and conversely.
Denote by té the set of all points belonging to infinitely many of the
intervals Ekii):
(4.15) 5f=\x\ xEC\EkAi,)},
j k,} and {¿„} being arbitrary infinite sequences. Consider any point, x,
ofS3; this point is the intersection of infinitely many Ek(i/), as follows
from the fact that the lengths of these intervals diminish to zero with in-
increasing A. The corresponding intervals Hkfjk) also intersect in a single
point y. For x£^, we now define \p(x) = y; the range of ^(x), for xE%?,
obviously is the set
3>= ¡y\ yEC\HKijk)\
It is readily verified that if x and x' belong to <ê, and x < x', then ^(x)
<iix'). It follows that the mapping ^: fé -+3> has a unique extension,
again denoted by t, on the positive real line, 3è+. This extension is mono-
tonic increasing and has the additional property
(4.16) Hx)EHkiJk)
whenever x E Ekii), jk being determined according to Lemma (i).
This completes the construction of the function ^(x). It only remains to
show that it belongs to class Lip[ln2/ln(2n + 2)]: we shall merely show
that this true for x E &• The argument below, however, easily extends to
any interval. We need
Lemma (ii). For any two points, xj¿x', in W, there exists a value of A for
which the two points are separated by at least one interval Ekii), or exactly
one gap, but by not more than 7 — 1 intervals.
Proof. For A = 1, there are at most 7 — 1 intervals Exii) between any
two points in Sf, because ^is covered by 7 + 1 of them (except, of course,
for the above-mentioned gaps). If two distinct points are never separated
by exactly one gap, or at most one interval, then they are separated by
N intervals Ekii) for some A. We recall that each Ekii) contains 7 — 1
intervals Ek+X(i'), and 7 — 2 gaps; each gap between the Ek(i) contains
one interval Ek+X(i') and two gaps (see Figure 1); the result follows from
the fact that the lengths of these diminish to zero with increasing A.
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 351
For given points, x?¿x', let now A be such that there are N intervals
EAi) and M gaps between them; according to Lemma (ii), TV = 1 if M = 0,
and M = 1 if/V=0, O^N^y-1 and O^M^y.Clearly,
(4.17) | x - x' | ^ 7V5* + Miy - 1) "V* > 2T~*-1;
at the same time, ^(x) and i^(x') must be contained in a span of at most
TV+2 intervals Hkijk) and M+2 gaps. Thus,
|fix) - *ix')\eiN+2)tk+iM + 2)ak,
where a* denotes the maximum gap between the Hkijk); we have noted
in the proof of Lemma (i) that ak ^ 2"*"1. Owing to (4.9) and the fact
that N^y-1,
iN+2)tk < (7 - D(t + 2)7H!*+1 < 2-*-1.
Consequently,
(4.18) |^(x)-^(x')| <2-*-1+(7 + 2)2-*-1=(7 + 3)2"*-1.
To obtain constants c and a for which
|*(x)-^(x')| f£c|x-xT,
we compare (4.17) with (4.18) and set
(7 + 3)2-*-1gc2"7-(*+1)o.
For c = 2"°(7 + 3), this inequality is valid if a = In2/ln7. Taking y = 2n
+ 2, as we may, we obtain the desired result. Clearly, better estimates
for c are possible by improving the estimates in (4.17) and those leading
to (4.18).
5. Proof of Lemma 1. For each A, we construct now 2n + 1 finite families,
SI, of pairwise disjoint sets, as demanded in Lemma 1; the members of
each family will, in fact, be cubes obtained as the cartesian products of
the intervals below.
Let ô > 0 be fixed; select A = A0 such that t = (7 — 1) ~ly~k ̂ 6; for each
q and A, define the intervals
(5.1) EUi) = [eAi) - eq,eAi) + 5k- tq],
obtained from the EAi) through the indicated translation to the left. Ob-
viously, Ekii) = Ekii) for each A, and being only a translation of the EAi),
the intervals (5.1) have the same intersection properties for increasing A
as the latter for each value of q.
For each q, let iq denote the restriction of i to the domain
(5.2) A!-{(7*-l)«g,...,7*+iy-lM;
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352 D.A.SPRECHER [March
for the remainder of this section we shall consider the intervals (5.1) only
for the values i = iq, and for A ̂ A0.
For i G A.Î, the EUi) have the covering property stated in
Lemma (iii). For fixed A, eacA point of & is contained in at least 2n inter-
vals EUi), 0^q^2n.
Proof. Let A be fixed. We observe that the points 0 and 1 are contained,
respectively, in the intervals corresponding to the values iq= iyk — l)tq
and iq = 7* + (7* — l)tq. Hence, it suffices to show that no two gaps from
those separating the intervals EUi) intersect as q and iq vary in their
respective domains.
The shift tq is an integral multiple of the gap-width between the EUi):
so is the length of the intervals. Consequently, two gaps will either coincide
or be mutually exclusive (except, possibly, for their end points). They will
overlap if and only if iqy~k — tq = i'qy~k — tq'; unless q = q' and i = i'
equality is impossible, owing to the inequality q — q' < y — 1.
n mû n ehù+d0iqí2n 0S«S2n
Figure 3
Lemma (iv). Let A be fixed. For each set of admitted values, i0, ••-, i^, given by
(5.3) iq=i0+il + tiyk-l))q,
the intersection of corresponding intervals is not empty:
(5.4) fi EUiq)^00g?S2n
for all values i0GA° and iq given by (5.3).
Proof. According to the foregoing, Ei+1iiq+x) will contain a gap separating
the Eiiiq) if and only if the initial point of the former coincides with the
terminal point of the latter (see Figure 3). Equating these points, we find that
iq+x=iq+l + tiyk-l),
relation (5.4) being obtained through an iteration. To verify (5.4), we have
only to show that eJfe(i2n) + ok — m — ekii0) > 0; we omit this simple
calculation.For each q and A, let iXq, • ■ •, in, be arbitrary values in Aj[; let the cor-
responding intervals, E?(iM), be laid on the pth coordinate axis of a rec-
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 353
tangular coordinate system in 3ën. The cartesian products
(5.5) S\(\)= n El(iK),lgpgB
in which i, denotes the vector i,= (¿i„ ••■,inq), define cubes in 3$n with
diameters y/n • bk, which, for fixed q and A, are separated by gaps of width
(7 — 1)_17~*; except for these, the cubes cover S3"1.
For each q and A, SI will denote the set of all cubes defined by (5.5):
(5.6) Si={S2(i,)| ¡MGA|).
We are now ready to complete the proof of Lemma 1. To establish the
asserted covering property, we shall show that every point of S£n is con-
tained in at least m + 1 cubes (5.5) as q varies in the domain 0 i£ q ^ n + m,
A being held fixed.
According to the properties of the intervals 2?j?(¿p,) already established,
each point, xp E -^p, appears for at least n + m of them. This being true
for each p, it follows that there are at most n values of q for which any
point, xE&n, is excluded from the cubes Sl(iq); this point belongs, there-
fore, to at least m + 1 of them (see Figure 4). That condition (1.5) is
satisfied follows from Lemma (iv); since the intersections (5.4) are non-
empty, neither are their cartesian products.
Clearly, &n can be expressed as the union of point sets generated from
infinite intersections of cubes Sl(iq) for each q.
6. Proof of Lemma 2 and Corollary 1. Throughout this section, q will vary
in the domain 0 ziq z^2n. We first show that the functions gg(x), as de-
fined by (1.6), satisfy Corollary 1 if they meet condition (1.7).
Let (a0, • • •, dfr) and (ß0, ■ • •, 02b) denote any permutations of the tuple
(0,1, ...,2n); let A be fixed. According to Lemma 1, each point of té"1 is
contained in at least n -\- 1 cubes S£(i„). Let the points x^ï' belong to cubes
Ski(iq) and S£«(iJ), respectively, for 0 á q zi n; on account of (1.7), the num-
bers g"«(x) and gMx') are all pairwise distinct. Thus, regardless of the
values of the remaining n pairs of images corresponding to n + 1 z% q z% 2n,
Corollary 1 is necessarily satisfied.
To prove (1.7), we begin with
Lemma (v). Let 0 < X < 1/n, then
supgqix) < inf^+1(x)
for all admitted values of q.
Proof. According to (4.16), Hx) EHkijk) whenever xEEkii); from
(4.14) we derive the fact that jki0) = 0 and ;*(7*) = yß for all A. Applied
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354 D.A.SPRECHER [March
to (4.11), this shows that ^(0) = 0 and tf-(l) = 1; because ^ is monotonie,
we have $& =të; it is easily verified that sup^ ^(x + «g) <3/2; (6.1) is
now obtained through a simple estimate.
It follows that (1.7) is valid at least for q j± r; hence, it remains only
to show that this relation is also valid when q = r. We point out that in
our present notation, i in (1.7) is replaced by iq.
Let us consider again the intervals EUi) for i E f; each EUi) is related
to EAi), in a one-to-one manner, through the equation
(6.1) x" = x - tq,
x"EEUi), xEEAi). It follows at once that
(6.2) tix + tq)EHkiJk)
for xEEUi)- For each p, q and A, define the intervals H^ijk) by
(6.3) HPUÙ = [hkijk) + q/n\", hkijk) + <* + q/n\p]
(see 4.10): Obviously,
(6.4) Hx + tq)+q/n\pEHPiJk)
whenever xEEUi). It is also clear that if we replace x by xp in (6.4) and
multiply each of these functions by Xp, then a summation over p will yield
the functions gqix), defined for all xp è 0.
For convenience of notation, we replace now jk by ;'. For each q, let jq
denote arbitrary admitted values of j corresponding to iq, where iq is re-
stricted to the domain A| (see 5.4); let j, = ijXq, •••,jnq), where the indices
j„ are admitted values of jq. Set
A2(j,)= Z MAjpq) + q/n\",ISpSn
(6.5) ^4=<k Z *",
ISpSn
and consider the intervals
(6.6) mUq) = [hU}q),Wi)q)+4].
The vectors j, being uniquely determined by the vectors i,, and their re-
lation being one-to-one for each q, it follows that, for fixed q and A, to each
cube, SUiq), there corresponds one, and only one, interval HU)q) under
the mapping gq; evidently,
(6.7) HU)q)=glSUiq).
We are now ready to prove (1.7) for the case r = q. The end points of
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1965] CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES 355
the intervals (6.6) do not coincide, because X satisfies no polynomial equa-
tion of degree less than n with integral coefficients. For fixed A, consider
then such an inequality as
hlih) > hlii'q);
our aim is to prove that
Af(j,) > huyq) + el
But, according to (4.7),
hli)q) - hlij'q) = y-*> ■ Z Wjpq-j'pq)>y-ek+i+1ISPSb
>7-%fi+i £ Xp>el,lgpgB
the last inequality owing to (4.9). This completes the proof of Theorem 1.
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